I notice that the integral of a linear function gives different results by using a method or another
Integrating directly,
\begin{equation} \int (a+bx) dx = ax+b\dfrac{x^2}{2} +c_1 \end{equation}
And making a substitution, \begin{equation} \begin{split} \int (a+bx) dx &\\ u=(a+bx) &\Rightarrow du=bdx \\ \int u \dfrac{du}{b} = \dfrac{1}{b}\dfrac{u^2}{2} +c_2 &=\dfrac{a^2}{2b}+\dfrac{bx^2}{2}+ax \end{split} \end{equation}
And comparing results, they are obviously not equal
\begin{equation} ax+b\dfrac{x^2}{2} +c_1 = ax +\dfrac{bx^2}{2} + \dfrac{a^2}{2b} +c_2 \end{equation}
My question is, how to explain that? Is it simply an arbitrary integration constants problem?